1 Introduction
The controller design presented is intended for the use in the field of process control. Normally in this field it is quite common for there to be found controllers based on the Proportionate Integral Derivative (PID) control method. PID control systems are generally used to control a nonlinear system around a predetermined operating point. In a properly designed PID control system, the design begins with the linearization of the nonlinear system around the operating point. The linearization is then followed by the preliminary selection of the respective gains Kp, Ki and Kd which are the proportionate, integral and derivative gains respectively based. During implementation, the control engineer is required to tune the values of these gains to acquire the desired closed loop response by viewing the trends of the process variable. Overtime the control engineer will be required to re-adjust the gains as the parameters of the system which were obtained from linearization have drifted away from the original values due to either the nonlinear nature of the process or a change in components of the process.
In most cases the drifting of parameters is the cause of changes in the response of the controller and therefore by extension a change in the quality of the product. For batch processes, maintaining consistency between batches is a requirement as there needs to be a small variation between batches of a given product. For a continuous process, this may result in more of the material produced having to be recycled back through the plant to obtain the required grade as in distillation systems. In addition to these concerns, the drifting of parameters can result in the system becoming unstable as the linearized close loop poles would have shifted to an unstable region. It is therefore the intention that the adaptive nonlinear control algorithm that is proposed address the issues that have been highlighted and provide the following features for nonlinear systems.
The features of the adaptive nonlinear control algorithm are:                1. Stability of the control system within the desired operating region.        2. A method to provide the system with the desired closed loop response.        3. Tracking of an input reference system.        
To demonstrate the useful nature of the control system with reference to the features described above, linear and nonlinear systems will be presented. The inclusion of the linear systems are to further illustrate the benefits and aid in the understanding of the control algorithm.
2 Background
As it was introduced in the previous section, practically all of the systems used in the process industry are nonlinear. Generally speaking all real systems exhibit some form of nonlinear behavior. Typical characteristics of a nonlinear system are the coupling of the state variables with each other or the coupling of the state variables with the control input. The aim of the invention is to provide an adaptive nonlinear controller capable of regulating the output of a nonlinear system. The obvious problem with the control of a nonlinear system is due to their nonlinearities. These nonlinearities can be attributed to the list of characteristics that are presented below with explanations. The invention provides a method to developing an adaptive nonlinear controller that can be applied to nonlinear systems in the presence of these problems.
I. Time-Varying Characteristics.                These characteristics relate to systems which are time variant and can be expressed in the simplest form in the equations below where A(t) is the time varying state matrix, x(t) is the state vector, B(t) is the input matrix, u(t) is the system input, y(t) is the output vector and C(t) is the output state matrix.{dot over (x)}(t)=A(t)x(t)+B(t)u(t)y(t)=C(t)x(t)        
II. Nonlinear Behavior                The nonlinear behavior of a system can be generated from the coupling of system inputs with the system states and also the inclusion of nonlinear functions within the state equations where x is the system state, u the system input, y the system output, f(x,u) a nonlinear vector of the system state and its input and h(x) is a nonlinear vector of the system state.{dot over (x)}=f(x,u)y=h(x)        
III. Model Inaccuracies                Model inaccuracies can arise from areas such as linearization and the assumptions made on the system parameters. Linearization of a nonlinear system can result in the system becoming ill-conditioned. For example using the system given above, the dynamics of a nonlinear system is linearized using the method shown in below to compute the linearized state equations.        
            A      ~        =                                                                      f                1                            ⁡                              (                                  x                  ,                  u                                )                                                    ∂                              x                1                                                              …                                                                    f                1                            ⁡                              (                                  x                  ,                  u                                )                                                    ∂                              x                n                                                                          ⋮                          ⋱                          ⋮                                                                ∂                                                f                  n                                ⁡                                  (                                      x                    ,                    u                                    )                                                                    ∂                              x                1                                                              …                                                                    f                n                            ⁡                              (                                  x                  ,                  u                                )                                                    ∂                              x                n                                                                    B      ~        =                                                                      f                n                            ⁡                              (                                  x                  ,                  u                                )                                                    ∂                              u                1                                                              …                                                                    f                1                            ⁡                              (                                  x                  ,                  u                                )                                                    ∂                              u                p                                                                          ⋮                          ⋱                          ⋮                                                                ∂                                                f                  n                                ⁡                                  (                                      x                    ,                    u                                    )                                                                    ∂                              u                1                                                              …                                                                    f                n                            ⁡                              (                                  x                  ,                  u                                )                                                    ∂                              u                p                                                        
The linearized dynamics of the nonlinear system are therefore∂{dot over (x)}=Ãδx+{tilde over (B)}δu. 
IV. Sensory feedback                The measurement of the output it important in feedback systems. In some cases all of the system states to be used in the controller are not available due to either the cost of equipment or a feasible method of measurement is not available. There is also the issue of the measurements being provided at the input of the controller after some time delay.3 Limitations of Present Solutions        
There are several limitations to the current control methods used to regulate nonlinear systems. The most common method is the application of linearization to the nonlinear system. The issue with linearization is that it restricts the control system developed to only being effective when all the system states are within a small region around the specified operating point. If the nonlinear system exits this region, the controller is no longer useful. One solution to this problem comes in the form of sliding mode control which links a series of local PID controllers at various regions of the systems operation. The problem with this solution is that there exist “chattering” when the system switches from one local controller to another local controller. This can cause oscillations within the system as the change in control action can send the system into a region where another controller has been specified and the action of the new controller returns the system to the other control region.
Model Predictive Control (MPC) is another method which can be used to control nonlinear systems. This control method is also based on the on initial development of a linearized model of the nonlinear system. Despite the known benefits of MPC, it is susceptible to an ill conditioned model. An Ill conditioned model occurs when there is a small variation of the linearized process parameters which cause the system poles of the linearized model to vary greatly.
PID controllers can be implemented and they assume that parameters of the linearized system are constant. Therefore for nonlinear systems, the response of the controller can vary and affect the performance of the closed loop system. Using a chemical process such as the distillation column as an example, the reduced effect of the control system can be explained. One of the purpose of control in a binary distillation column is to regulate the quality of the composition of the product streams exiting the column. Over time the parameters vary due to changes in the characteristics of the pumps and the internal flows within the column. PID controllers are developed using some knowledge of the system to place the system at specific operating points. Therefore changes in the parameters over time cause the poles to be shifted to regions which may be undesirable. If we consider the shifting of the closed loop system poles, the system response may acquire larger oscillations and take a longer time to settle to the reference values.
Another solution to the control of nonlinear systems is feedback linearization. This control strategy removes the nonlinearities of the system through feedback given that the nonlinearities are all located within the same state equation as the control input. If there are any nonlinearities within other state equations, transformation of the system to a controllable form is therefore required. To successfully apply this technique all the parameters of the system must be known. An additional drawback of this type of control is that it cannot be applied to systems where the coefficient of the control input approaches zero and therefore causes a singularity at the control input.
It should be understood that there do exist control strategies which are capable of controlling nonlinear processes as shown in the referenced documents. However, they lack certain characteristic features of the algorithm which is being proposed. What is needed are non-linear control systems and methods that address the foregoing problems.
The invention provides systems and methods for generating an adaptive nonlinear controller and for applying the adaptive nonlinear controller to regulating nonlinear process systems in the presence of the foregoing problems in a manner that minimizes the foregoing problems. The next section will illustrate the features of the adaptive non-linear control algorithm which make it unique and demonstrate its development and implementation.